AKG's point is that N is NOT a function from the real numbers to the real numbers. It is a function whose domain is a set of functions, and whose range is the real numbers. How are you defining the derivative of such a function?

I actually don't know. I just know that in a general metric space, a function f is differentiable at one point y if it can be written as f(x)=f(y)+df(x-y)+p(x) where p is little-o norm and df(x-y) is a linear continous function. I suspect df is the derivative, but I don't know how to compute it.

There is no mentioning of differentiation in my book (Elements of the Theory of Functions and Functional Analysis) and I might have made a mistake in my notes.

I actually don't know. I just know that in a general metric space, a function f is differentiable at one point y if it can be written as f(x)=f(y)+df(x-y)+p(x) where p is little-o norm and df(x-y) is a linear continous function. I suspect df is the derivative, but I don't know how to compute it.

There is no mentioning of differentiation in my book (Elements of the Theory of Functions and Functional Analysis) and I might have made a mistake in my notes.

In a "general metric space", addition is not defined. What you are talking about is a topological vector space (the most general situation in which the derivative can be defined). Sets of functions do form topological vector spaces in which the topology is based on a metric but there are several different ones that are not equivalent. You haven't said which metric you are using.

I actually don't know. I just know that in a general metric space, a function f is differentiable at one point y if it can be written as f(x)=f(y)+df(x-y)+p(x) where p is little-o norm and df(x-y) is a linear continous function. I suspect df is the derivative, but I don't know how to compute it.

There is no mentioning of differentiation in my book (Elements of the Theory of Functions and Functional Analysis) and I might have made a mistake in my notes.

Your definition of little-o norm depends on x0, i.e. it could only be read to say that f is little-o norm at x0. In your definition of differentiability, then, do you mean p to be little-o norm at some point, or at all points?

Also, the only way your definition makes sense is if you mean df to be linear continuous, not df(x-y).

The definition that would make the most sense to me is that given f : V -> W, f is differentiable at some y in V iff there exists a function p : V -> W that is little-o norm at y and a function df(y) : V -> W that is continuous and linear such that for all x in V, f(x) = f(y) + (df(y))(x-y) + p(x). Note that if V = W = R, then a continuous linear function df(y) : V -> W just multiplies its argument by a constant. So if f'(y) is what we think of as being the derivative of f at y in the normal sense (i.e. it is just a number), then df(y)(z) = f'(y)z, where on the left we have the function df(y) acting on z, and on the right we have a number f'(y) multiplying with z.

Check that p so-defined is little-o norm at f. Guessing dN(f) is the tricky part, because as far as I know, you just have to make a reasonable guess, there's no smart way I know of of picking the right dN(f). This might be because this is the first time I'm seeing differentiation on an arbitrary normed vector space. If you pick the right dN(f), then showing p is little-o norm at f is just a first-year analysis sort of [itex]\delta -\epsilon[/itex] problem.